1. Field of the Invention
The present invention relates to a method of modeling and analyzing electronic noise and, more particularly, to a method of obtaining a frequency-dependent analytical Pade approximation of the noise power spectral density of an electronic circuit or system.
2. Description of the Related Art
Noise is a fundamental phenomenon in electronic circuits, caused by the small fluctuations in currents and voltages that occur within the devices of the electronic circuit. The fluctuations are due mainly to the discontinuous nature of electric charge. Determining the effects of noise on electronic circuits and systems is important, as noise often represents the fundamental practical limit of circuit or system performance.
Noise analysis algorithms for circuits in DC steady-state have long been available in commercial simulation software programs such, for example, as SPICE. Such programs typically calculate noise power at a certain user-designated frequency or range of frequencies and provide the calculated data in tabulated form. Circuit or system designers generally reduce the information contained in the noise spectrum, as calculated by the modeling program, to a single number that represents the noise figure for the circuit or system. Such compact representations offer good insight into the circuit or system performance and are convenient for quick estimation of the effect of noise on the circuit or system. However, computer-aided design (CAD) tools for both the circuit and system levels can take advantage of the more accurate and complete information available in the noise spectrum, which is not present in the compact representation provided by the single noise figure. Furthermore, the complete frequency-dependent noise spectrum information calculated by known programs would require hundreds and possibly thousands of iterations to analytically fully model the noise power spectral density of the circuit or system.
The principal sources of noise in integrated circuits are: thermal noise, which occurs in almost all devices as a result of the thermal agitation of the electrons; shot noise, which is associated with direct current flow mainly in bipolar diodes and transistors and is typically due to the fact that the current through a junction consists of discrete charge carriers randomly crossing a potential barrier; and flicker (or 1/f) noise, which occurs in all active devices, and even in some resistors, and is also associated with direct current flow.
Mathematically, integrated circuit device noise is modeled by stochastic processes. A noise stochastic process is a function of time n(t), the value of which at each time point is a random variable. Stochastic processes are characterized in terms of statistical averages, such as the mean and autocorrelation in the time domain, and the power spectral density in the frequency domain.
For example, the thermal noise of a resistor is modeled by a current source in parallel with the resistor. The value of the current source is a zero-mean stochastic process with a constant spectral density at all frequencies equal to EQU S.sub.th (.omega.)=4kTG, (1)
where k is Boltzman's constant, T is the absolute temperature, and G is the conductance. Such a process (i.e. a zero-mean stochastic process), having a spectral density not dependent on frequency, represents white noise.
Shot noise in a junction is also modeled by a white noise current source in parallel with the junction. The spectral density of shot noise is characterized by EQU S.sub.sh (.omega.)=2qI.sub.d, (2)
where q is the electron charge and I.sub.d is the average current through the junction.
Flicker noise is modeled by a stochastic process with a non-constant spectral density according to the following equation: ##EQU1## where I is the average direct current, K.sub.I is a constant for a particular device and process, a is a constant in the range of 0.5 to 2.0, and b is a constant of about one; hence the name 1/f noise. The circuit equations that include the noise excitation are ##EQU2## Here, x(t) is the vector of circuit variables, typically currents and voltages, f(x(t)) represents the contribution of the resistive components, q(x(t)) is the contribution of the reactive components, b.sub.0 is the constant (DC) excitation, B is the noise-source incidence matrix, and n(t) is a vector stochastic process that describes the noise sources. The vector stochastic process n(t) is specified in terms of its frequency-domain cross-spectral density matrix represented by S.sub.xx (.omega.). The diagonal elements in S.sub.xx (.omega.) represent the power spectral density of each noise source, and the off-diagonal elements describe statistical coupling of noise signals. In most cases, the noise sources model mutually independent phenomena, hence the corresponding noise sources will be uncorrelated and all off-diagonal elements in S.sub.xx will be zero. In practical cases, therefore, S.sub.xx will almost always be a diagonal matrix. The model, however, is sufficiently general to capture correlated noise sources when necessary; in this case, non-zero off-diagonal elements will exist in S.sub.xx.
Assuming that x.sub.0 is the solution of the noiseless DC circuit, constant in time, ##EQU3## The response of a circuit in the presence of noise will be a perturbation, z(t), of the DC solution, x.sub.0, ##EQU4## Assuming that the noise signals are small relative to the other signals present in the electronic circuit, the first-order Taylor expansion of equation (6) for the DC solution is sufficiently accurate to model and analyze the circuit in terms of noise as follows: ##EQU5## Considering equation (5) and the fact that q(x.sub.0) and .differential.f/.differential.x.vertline..sub.x.sbsb.0 are constants in time, the linear, stochastic, differential equation for the noise signals represented by the following equation remains ##EQU6##
Thus, the noise-analysis problem reduces to that of the propagation of a stochastic process through a linear system.
The general expression of the noise power spectral density at the output of the linear system, S.sub.yy, is given by the well known formula EQU S.sub.yy (.omega.)=H(j.omega.)S.sub.xx (.omega.)H.sup.H (j.omega.).(9)
Note that S.sub.yy (.omega.) is a scalar function of frequency. For analysis of more than one output, S.sub.yy (.omega.) is the noise power cross-spectral density matrix, a full square matrix, the dimension of which is the number of outputs. In this case, the diagonal elements of S.sub.yy (.omega.) represent the power spectral density of the noise at each output and the off-diagonal elements represent the cross-spectral densities of output pairs.
For a single-output system, the many-to-one vector transfer function of the linear system from the noise sources to an output port of interest is EQU H(j.omega.)=I.sup.T (G+j.omega.C).sup.-1 B, (10)
where l denotes the incidence vector that corresponds to the output port of interest. More generally, and for a multi-output system, with p denoting the number of output ports (i.e. when p is greater than 1), there exists a many-to-many matrix-transfer-function from noise sources to the p outputs, defined by EQU H(j.omega.)=L.sup.T (G+j.omega.C).sup.-1 B, (11)
where L is the incidence matrix (with p columns) of the output ports. Each column of L is an incidence vector corresponding to one of the ports. From the relationship of equation (9), and using equation (11), the noise power cross-spectral density matrix at the output of the system may be obtained using the following equation EQU S.sub.yy (.omega.)=L.sup.T (G+j.omega.C).sup.-1 BS.sub.xx (.omega.)B.sup.T (G+j.omega.C).sup.-H L. (12)
The noise modeling and analysis method implemented in prior art computer or calculation programs such for example as SPICE, evaluates this expression (i.e. equation (12)) efficiently, for a given value of .omega.), using the solution of the adjoint system EQU S.sub.yy (.omega.)=x.sup.H.sub.a (j.omega.)BS.sub.xx (.omega.)B.sup.T x.sub.a (j.omega.), (13)
where x.sub.a (j.omega.) is the solution of the adjoint system EQU x.sub.a (j.omega.)=(G+j.omega.C).sup.-H L
Noise modeling and analysis using the above-described prior art methods require that frequency domain equations be solved at each discrete frequency point of interest. Depending on the complexity of the circuit or system, this may require hundreds or thousands of computations--the result of which is a table comprised of noise values for each of the desired frequency points. In addition to the disadvantageous expense of these prior art methods by virtue of the time required to model complex circuits and systems, the results produced are not amenable for hierarchical use.